Karl Dilcher and I have made the first extension of the Gauss-Wilson theorem since the appearance of Gauss' Disquisitiones. Defining N_n! - the 'Gauss factorial' of N with respect to n - to be the product of the residue classes in [1, N] that are relatively prime to n, we have given a complete determination of the order of ((n-1)/2)_n! mod n. This is a composite modulus extension of Mordell's 1961 result concerning the order of ((p-1)/2)! mod p (for prime p).
I shall outline work-in-progress concerning the order of ((n-1)/M)_n! mod n for M = 3 and 4, introduce a new class of primes (Gauss-4 primes), and outline a number of open problems.
John B. Cosgrave